Cartan if and only if it is a maximal toral subalgebra. An advantage of this approach is that it is introduction to kac-moody algebra pdf to show the existence of such a subalgebra.
Any nilpotent Lie algebra is its own Cartan subalgebra. 2 by 2 matrices of trace 0 has two non-conjugate Cartan subalgebras. The dimension of a Cartan subalgebra is not in general the maximal dimension of an abelian subalgebra, even for complex simple Lie algebras. Over non-algebraically closed fields, not all Cartan subalgebras are conjugate. Lie algebras over algebraically closed fields.
Over a non-algebraically closed field not every semisimple Lie algebra is splittable, however. This page was last edited on 5 July 2017, at 18:28. The relevant automorphism can be associated with parity. Check if you have access through your login credentials or your institution.